# Tagged Questions

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### Intuitive meaning of the exponential form of an unitary operator

I'm an undergraduate student in Chemistry currently studying quantum mechanics and I have a problem with unitary transformations. Here in my book, it is stated that Every unitary operator ...
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### Shrinkage operator for matrices

Here http://web.stanford.edu/~boyd/papers/pdf/prox_algs.pdf, on page 188, you can see the derivation of the soft thresholding operator or shrinkage operator for the case of vectors using Moreau ...
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### What is the closest self-adjoint (positive) operator to a given operator?

Given an operator $\rho$ on a Hilbert space $H$, is there a notion of nearest self-adjoint (positive) approximation of $\rho$ for a suitable norm? More specifically, does there exist an algebraic ...
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### Canonical Forms For Matrices

In the following paper by Wedderburn what are the restrictions on the field $\mathbb F$ or on the linear application $\varphi$ that the author refers to obtain the matrix B? ...
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### Relation between $A^{*}B=B^{*}A$ and $AB^{*}=BA^{*}$

Let $A$ and $B$ be two matrices. Can we say $A^{*}B=B^{*}A$ implies $AB^{*}=BA^{*}$? how about when $A$ or $B$ are normal? Any comments could be useful. Thanks.
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### norm of a nilpotent matrix

A proof I was reading used the claim that $||{N}||_2$ = 1 for a nilpotent matrix $N$. I tried to prove it, and have a couple of questions on it. First, my "proof": We know that there exists a basis ...
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### Norm of the multiplication operator

Let $f \in L^\infty[0,1].$ It is clear that the norm of the multiplication operator $M_f : g \mapsto fg$ on $L^p[0,1]$ is $\|f\|_\infty.$ What happens in the noncommutative situation? Let us ...
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### Find a real matrix with eigen vectors v and v's complex conjugate so that they have different eigenvalues.

I need to find a real matrix with eigenvector v, and eigenvector v's complex conjugate, such that they will have different eigenvalue. any hints please?
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### $T:V\rightarrow V$ is over $\mathbb{R}$ , it's matrix is $A$, $A=PDP^*$. Is it true that $A$, $D$, and $P$ are in $M_{n \times n}(\mathbb{R})$

$T:V\rightarrow V$ is over $\mathbb{R}$ and $V$ of finite dimension $n$, and I know that it is orthogonally diagonalizable. The Matrix that represents it - call it $A$ ,in orthonormal basis is ...
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### $T^*T=TT^*$ and $T^2=T$. Prove $T$ is self adjoint: $T=T^*$ [duplicate]

$V$ is an inner product space of finite dimension over $\mathbb{R}$, and $T:V\to V$ a linear transformation which is normal, that is, $T^*T=TT^*$. In addition $T^2=T$. Prove $T$ is self adjoint, that ...
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### If A unitary matrix and orthogonally diagonalizable why there is a basis in whichthe linear trans. matrix is diagonal?

If $A$ is a $n\times n$ unitary matrix (above the complex field) and is orthogonally diagonalizable, why does it mean that the is an orthonormal basis $\mathbb C$ in which the matrix that represent ...
470 views

### Expectation Operator on a Matrix

Kind of embarrassing, but I'm completely blanking on what applying the expectation operator to a matrix means, and I can't find a simple explanation anywhere, or an example of how to carry out the ...
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### relations between two linear operators

Let $\alpha,\beta$ be linear operators on a finite dimensional vector space $V$ over field $F$. Let $\gamma=\alpha\circ\beta$ and $\delta=\beta\circ\alpha$. Prove that: (1). $m_\delta(x)$ divides ...
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### Prove that if $T$ a normal linear transformation and invertible, then $T^{-1}$ is normal.

The question is: Prove that if $T$ a normal linear transformation and invertible, then $T^{-1}$ is normal. Then I have to find the spectral decomposition of $T^{-1}$. At first I tried to prove it by ...
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### Why is this true: The only orthogonal projection that is also unitary from $\Bbb C^n$ to $\Bbb C^n$ is the identity

Can anyone explain me please how to see this statement: the only orthogonal projection that is also unitary from $\Bbb C^n$ to $\Bbb C^n$ is the Identity. how can I prove formally that? or how can I ...
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### find the eigenbasis of unitary transformation

$U$ is $n\times n$ unitary matrix, with orthogonal eigenbasis $v_1, \ldots v_n$ we construct a linear transformation: $T_U(X) = XU$ with the inner product $\langle A, B \rangle = \text{tr}(A^*B)$ I ...
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### Question arising from quantum mechanics concerning groups and symmetries

I'm trying to understand a calculation my professor did in my quantum mechanics script. Here it is: Each rotation $R \in O(3)$ induces a unitary transformation in $L^2(R^3)$, i.e. the space of square ...
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### Is it true that every orthogonal transformation , even over $\mathbb R$, is diagonalizable?

Is it true that every orthogonal transformation , even over $\mathbb R$, is diagonalizable? I didn't succeed to get any information about it. Could anyone explain please?
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### Sum of the matrix series

Let $A\in\mathbb R^{n\times n}$ be a symmetric matrix which $0\preceq A\preceq I$ ($I$ is identity matrix), and $w_k\in\mathbb R^n$ are arbitrary certain vectors which $\|w_k\|\leq1,\,\,k=0,1,\ldots$ ...
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### Prove that if transformation matrix is unitary, then the basis is orthonormal

V is a vector space with the complex field, B is an orthonormal basis of V , and C is some arbitrary basis. Prove that if the transformation matrix from basis C to B is unitary, then C is also ...
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### Compute the norm of matrix

Let $M$ be $n\times n$ matrix, consisting entirely of 1's. Show, that $\|M\|_{op}=\sup_{x\in C^n}|Mx|=n$.
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### Notation for Kronecker product of a matrix and itself?

What is the notation for the Kronecker product of a matrix and itself? In other words, is there a short-hand way I can express the following: $X⊗X$ $X⊗X⊗X$ $X⊗X⊗X⊗X$ Where $X$ is a matrix? What ...
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### Relation of norms of matrices

Let $A$ be $m \times n$ matrix. Let $B=\frac 1n AA^*$, where $A^*$ is a transposed matrix. Let $X_i, I\leq m$ be row-vectors of $A$. Show $$\|B\|=\frac 1n \|A\|^2\geq \max_{i\leq m}|X_i|,$$ Where, ...
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### bounding operator norm by quadratic forms on orthonormal basis

Let $\{u_i\}_{i=1}^n$ be a set of orthonormal basis and $M$ a symmetric matrix. Suppose $$\left|\langle u_iu_i^T,M\rangle\right|\leq \tau\ \ \forall i=1,\dots,n.$$ Can I ...
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### Operators that are not represented as matrices , operating on matrices.

I am currently going through "Log-gases and random matrices" by PJ Forrester. I'm coming from a totally different academic background, and I cannot understand a point of his notation. More precisely, ...
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### Magnitude of a Matrix?

Consider a vector V. The magnitude of this vector (if it describes a position in euclidean space) = distance from the origin is simply: $(V^TV)^{1/2}$ aka the square root of the dot product... ...
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### limit of evaluated automorphisms in a Banach algebra

Let $\mathcal{A}=\operatorname{M}_k(\mathbb{R})$ be the Banach algebra of $k\times k$ real matrices and let $(U_n)_{n\in\mathbb{N}}\subset\operatorname{GL}_k(\mathbb{R})$ be a sequence of invertible ...
398 views

### Matrix norm of a normal matrix

A normal matrix defined over a complex vector space has the property, that $\|A\|_2$ is its largest eigenvalue and now I was wondering whether this is also true for matrices defined over the real ...
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### Trace of a differential operator

Given the differential operator: $$A=\exp(-\beta H)$$ where $$H=\frac{1}{2}\left( -\frac{d^2}{dx^2}+x^2 \right)$$ and $\beta\gt 0$ How can I get the trace of this operator? Thanks in advance.
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### $(P\Lambda P^{-1}=T^2)~\implies~(\exists \Lambda'~\text{s.t.}~T=R\Lambda' R^{-1})$: $\;P,R\;$ Unitary Matrices

Let $T$ be a linear operator such that the operator $T^2$ is diagonalizable. Is $T$ necessarily diagonalizable?
Consider a linear operator on $\mathbf{R}^n$ represented by a square matrix of size $n \times n$, call it $A$. The matrix acts on a row vector, call it $x$ and returns a row vector, call it $x'$, so ...